Ecological Modelling 171 (2004) 21–33 Bottom-up and top-down effects in food chains depend on functional dependence: an explicit framework Robert A. Herendeen∗ Illinois Natural History Survey, Champaign, IL 61820 USA Received 5 July 2002; received in revised form 14 May 2003; accepted 2 June 2003 Abstract Observed stock changes in perturbed ecosystems sometimes, but not always, are smaller than predicted by the trophic cascade hypothesis. These varying outcomes can be explained by (1) using detailed analysis of trophic-level interactions within the standard energy-based linear food-chain model, or (2) invoking web models and/or non-energy interactions between organisms. Previously I developed an analytic approach for the linear chain for a press-type perturbation and applied it to ratio-dependent functional relationships. Here I extend the linear chain analysis to a more general functional relationship which allows independent variation of prey dependence and intra-level interference. I find that different combinations of prey dependence and interference lead to large or small cascading effects. Generally, large top-down effects require weak interference, while large bottom-up effects require both weak interference and strong prey dependence. © 2003 Elsevier B.V. All rights reserved. Keywords: Trophic cascade; Top-down; Bottom-up; Food chain; Ratio dependent; Prey dependent; Predator dependent 1. Introduction ing a high trophic level becomes undetectable two trophic levels down the chain. In Herendeen (1995) An ecosystem is said to exhibit a trophic cascade I argued, using an analytical model and simulations, (TC) when perturbing the stock of a higher trophic that for a press perturbation, this diminution is to be level results in observable changes in the stocks of expected for ratio-dependent predator–prey relation- lower trophic levels. The mirror image, i.e. conse- ships. In this paper I extend chain analysis for a press quences of perturbing a lower trophic level, is called perturbation to incorporate variable prey dependence the bottom-up effect. The trophic cascade is often seen and interference in each trophic level. Response to a experimentally, but often it is not. Many reasons for periodic perturbation will be covered in a subsequent its absence derive from various manifestations of food article (Herendeen, in preparation). webs rather than linear chains, but it is not necessary I will show that this approach predicts large or small to abandon chains to explain a wide range of obser- TCs depending on the degree of prey dependence and vations. A typical result is that the effect of perturb- intralevel interference. The paper is organized as fol- lows. ∗ Tel.: +1-217-244-2137; fax: +1-217-333-6294. Section 2: Background on trophic cascades seen and E-mail address: [email protected] (R.A. Herendeen). not seen. 0304-3800/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-3800(03)00273-4 22 R.A. Herendeen / Ecological Modelling 171 (2004) 21–33 Section 3: General analytical approach for a Lawler, 1995; Chase, 1996; Moran et al., press-perturbed food chain. 1996; Schmitz et al., 2000; Beckerman et al., Section 4: Response of a 3-level food chain to a 1997; Pace et al., 1998; Turchin et al., 2000). press perturbation. Specific issues are refuges, prey-dependent Section 5: Obtaining large top-down and small instead of ratio-dependent predation (one ex- bottom-up effects. ample being Lotka–Volterra dynamics, which Section 6: Conclusions. tends to produce oscillations), and intrat- rophic level interference (Rosenheim et al., 1993; McCann et al., 1998). 2. Background: trophic cascades seen and not seen Experimental: Experiments are inadequately defined and ex- The trophic cascade hypothesis has often been ecuted regarding temporal behavior: criticized because the observed effects are smaller (a) The time profile of the perturbation and than one hopes for (Diana et al., 1991; Baca and the expected response is ambiguous: is the Drenner, 1995; Carter and Rypstra, 1995; Brett and perturbation a pulse, a press (a step func- Goldman, 1996, 1997; Brönmark and Weisner, 1996; tion that persists indefinitely), or periodic, Mullersolger et al., 1997; Mikola and Setälä, 1998; such as a sinusoid in time (Blaustein et al., Bertolo et al., 2000). That is, the change in the stock 1995; Leibold et al., 1997)? If top-down and of one trophic level is less than expected, often to the bottom-up perturbations are used simultane- point of undetectability, when the stock of another ously, is there clear delineation between the trophic level is changed. Because of early claims that two (Diana et al., 1991)? the trophic cascade would be a powerful management (b) Experiments are not run long enough (often tool (e.g. to control aquatic vegetation by manipu- for good and practical reasons) for transient lating fish populations), this has led to a number of effects to damp out (Leibold et al., 1997; criticisms (DeMelo et al., 1992). These include: Persson, 1997; Pace et al., 1998; Polis et al., 2000). Conceptual: 1. Real ecosystems are webs, not chains as the With all these objections there is still an argument TC assumes (Hill and Lodge, 1995; Polis and for chain-like trophic effects. Hairston and Hairston Strong, 1996; Polis et al., 2000). Recent work (1997) claim that even though omnivory is more likely has shown the strength of several mecha- in terrestrial than in aquatic systems, trophic-level nisms that work against the TC, for example, dynamics is still often observed. This harkens back omnivory (Nyström et al., 1996; Charlebois to Hairston et al.’s (1960) three-level “green world” and Lamberti, 1996; Strong, 1999), and nu- hypothesis. trient loops (Carpenter et al., 1992; Findlay In addition, the size of the sought-after TC effect et al., 1994; Vanni and Layne, 1997; Vanni has often not been carefully predicted. Implicitly, one et al., 1997; Perez-Fuentetaja et al., 1996). is seeking effects in distant trophic levels of a magni- Polis (1999) argued that chain-like dynamics tude comparable with the perturbed level. For exam- is much more likely (for trophic levels as dis- ple, halving the biomass of piscivorous fish is casually tinct from individual species) in aquatic than expected to produce roughly a doubling or halving of in terrestrial systems. stocks in other trophic levels. While effects of this 2. Even with chain structure, shifts in the magnitude are sometimes seen (Marquis and Whelan, strength and functional dependence of preda- 1994; Wootton, 1995; Chase, 1996; Moran et al., 1996; tion, as well as non-energy behavioral inter- Moran and Hurd, 1998; Estes et al., 1998; Nicholls, actions, change the quantitative interactions 1999; Schmitz et al., 2000), the mechanism of the TC of trophic levels and lead to responses not does not require them. predicted by the simple (time independent) It is not necessary to invoke the above concep- relations assumed in the TC (Balciunas and tual criticisms and abandon a time-independent linear R.A. Herendeen / Ecological Modelling 171 (2004) 21–33 23 chain model to explain finding small or no TC effects. 3. General analytical approach for a In Herendeen (1995) I showed analytically, with sim- press-perturbed food chain ulation corroboration, that for ratio-dependent preda- tion, the strength of the TC can fall off rapidly down Assume a food chain of k trophic levels, as shown the chain. Combining this result with the experimen- in Fig. 1.(Table 1 contains definitions of all symbols.) tal criticism above, we then have the possibility that For each level i, the general, steady state biomass researchers are often looking for too-large effects in energy conservation equation is: too-quick experiments, and, not surprisingly, often not INPUTi = METMORTi + CROPPINGi + INPUTi+ finding them. 1 In Herendeen (1995) I investigated the case in (1) which an one level in a chain is perturbed (via a where INPUTi is the energy flow into level i resulting positive or negative step function change in cropping from preying upon level i − 1, METMORTi is the or in light level) so that the level eventually settles energy flow out of level i resulting from metabolism down to a fractional stock change of 1 unit (arbitrarily and non-predation mortality, CROPPINGi is the en- chosen). This is the experiment envisioned, if not ex- ergy flow out of level i resulting from cropping plicitly articulated or achieved, by many researchers. (if negative, it represents stocking), INPUTi+ is I found that with ratio-dependent predation, the frac- 1 the energy flow out of level i resulting from pre- tional stock change diminishes down the trophic dation by level i + 1, Si is the energy stock in chain, but is approximately the same up the trophic level i. chain. The method allowed perturbing several levels METMORT is assumed to be proportional to stock simultaneously, which covers experiments combining (i.e. METMORTi = µiSi). Bi(Si,Si− ) is the input simultaneous top-down and bottom-up manipulations 1 per unit stock of level i, where Bi depends nonlinearly such as reported by McCarty (1997) and Carpenter on the stocks of predator and prey. For the lowest et al. (1996). One early hope was that manipulating trophic level the “prey” is light and nutrients, which I top carnivores in eutrophicated lakes would cascade to call RESOURCE. I will consider three types of press control algal blooms, but the latter authors concluded perturbation: that “... the potential for increasing eutrophication [of a lake system] by P[hosphorus] input exceeds the 1. Changes in CROPPING. potential for controlling eutrophication by food web 2. Changes in RESOURCE. manipulation” (i.e. bottom-up manipulation is more 3. Changes in Bi(Si,Si−1), i.e. in functional depen- effective than top-down).